Table 34.1.1–1 Available loads and predefined fields.

Loads and predefined fields	Procedures
Added mass (concentrated and distributed)	Abaqus/Aqua eigenfrequency extraction analysis (“Natural frequency extraction,” Section 6.3.5)
Base motion	Procedures based on eigenmodes: “Transient modal dynamic analysis,” Section 6.3.7 “Mode-based steady-state dynamic analysis,” Section 6.3.8 “Response spectrum analysis,” Section 6.3.10 “Random response analysis,” Section 6.3.11
Boundary condition with a nonzero prescribed boundary	All procedures except those based on eigenmodes
Connector motion Connector load	All relevant procedures except modal extraction, buckling, those based on eigenmodes, and direct steady-state dynamics
Cross-correlation property	“Random response analysis,” Section 6.3.11
Current density (concentrated and distributed)	“Coupled thermal-electrical analysis,” Section 6.7.3 “Fully coupled thermal-electrical-structural analysis,” Section 6.7.4
Current density vector	“Eddy current analysis,” Section 6.7.5
Electric charge (concentrated and distributed)	“Piezoelectric analysis,” Section 6.7.2
Equivalent pressure stress	“Mass diffusion analysis,” Section 6.9.1
Film coefficient and associated sink temperature	All procedures involving temperature degrees of freedom
Fluid flux	Analysis involving hydrostatic fluid elements
Fluid mass flow rate	Analysis involving convective heat transfer elements
Flux (concentrated and distributed)	All procedures involving temperature degrees of freedom “Mass diffusion analysis,” Section 6.9.1
Force and moment (concentrated and distributed)	All procedures with displacement degrees of freedom except response spectrum

Loads and predefined fields	Procedures
Incident wave loading	Direct-integration dynamic analysis (“Implicit dynamic analysis using direct integration,” Section 6.3.2) involving solid and/or fluid elements undergoing shock loading
Predefined field variable	All procedures except those based on eigenmodes
Seepage coefficient and associated sink pore pressure Distributed seepage flow	“Coupled pore fluid diffusion and stress analysis,” Section 6.8.1
Substructure load	All procedures involving the use of substructures
Temperature as a predefined field	All procedures except adiabatic analysis, mode-based procedures, and procedures involving temperature degrees of freedom

With the exception of concentrated added mass and distributed added mass, no loads can be applied in eigenfrequency extraction analysis. # 34.1.2 AMPLITUDE CURVES Products: Abaqus/Standard Abaqus/Explicit Abaqus/CFD Abaqus/CAE # References • “Prescribed conditions: overview,” Section 34.1.1 • \*AMPLITUDE • Chapter 57, “The Amplitude toolset,” of the Abaqus/CAE User’s Guide # Overview An amplitude curve: • allows arbitrary time (or frequency) variations of load, displacement, and other prescribed variables to be given throughout a step (using step time) or throughout the analysis (using total time); • can be defined as a mathematical function (such as a sinusoidal variation), as a series of values at points in time (such as a digitized acceleration-time record from an earthquake), as a user-customized definition via user subroutines, or, in Abaqus/Standard, as values calculated based on a solution-dependent variable (such as the maximum creep strain rate in a superplastic forming problem); and • can be referred to by name by any number of boundary conditions, loads, and predefined fields. # Amplitude curves By default, the values of loads, boundary conditions, and predefined fields either change linearly with time throughout the step (ramp function) or they are applied immediately and remain constant throughout the step (step function)—see “Defining an analysis,” Section 6.1.2. Many problems require a more elaborate definition, however. For example, different amplitude curves can be used to specify time variations for different loadings. One common example is the combination of thermal and mechanical load transients: usually the temperatures and mechanical loads have different time variations during the step. Different amplitude curves can be used to specify each of these time variations. Other examples include dynamic analysis under earthquake loading, where an amplitude curve can be used to specify the variation of acceleration with time, and underwater shock analysis, where an amplitude curve is used to specify the incident pressure profile. Amplitudes are defined as model data (i.e., they are not step dependent). Each amplitude curve must be named; this name is then referred to from the load, boundary condition, or predefined field definition (see “Prescribed conditions: overview,” Section 34.1.1). Input File Usage: \*AMPLITUDE, NAME=name Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: Name: name # Defining the time period Each amplitude curve is a function of time or frequency. Amplitudes defined as functions of frequency are used in “Direct-solution steady-state dynamic analysis,” Section 6.3.4, “Mode-based steady-state dynamic analysis,” Section 6.3.8, and “Eddy current analysis,” Section 6.7.5. Amplitudes defined as functions of time can be given in terms of step time (default) or in terms of total time. These time measures are defined in “Conventions,” Section 1.2.2. Input File Usage: Use one of the following options: \*AMPLITUDE, NAME=name, TIME=STEP TIME (default) \*AMPLITUDE, NAME=name, TIME=TOTAL TIME Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: any type: Time span: Step time or Total time # Continuation of an amplitude reference in subsequent steps If a boundary condition, load, or predefined field refers to an amplitude curve and the prescribed condition is not redefined in subsequent steps, the following rules apply: • If the associated amplitude was given in terms of total time, the prescribed condition continues to follow the amplitude definition. • If no associated amplitude was given or if the amplitude was given in terms of step time, the prescribed condition remains constant at the magnitude associated with the end of the previous step. # Specifying relative or absolute data You can choose between specifying relative or absolute magnitudes for an amplitude curve. # Relative data By default, you give the amplitude magnitude as a multiple (fraction) of the reference magnitude given in the prescribed condition definition. This method is especially useful when the same variation applies to different load types. Input File Usage: \*AMPLITUDE, NAME=name, VALUE=RELATIVE Abaqus/CAE Usage: Amplitude magnitudes are always relative in Abaqus/CAE. # Absolute data Alternatively, you can give absolute magnitudes directly. When this method is used, the values given in the prescribed condition definitions will be ignored. Absolute amplitude values should generally not be used to define temperatures or predefined field variables for nodes attached to beam or shell elements as values at the reference surface together with the gradient or gradients across the section (default cross-section definition; see “Using a beam section integrated during the analysis to define the section behavior,” Section 29.3.6, and “Using a shell section integrated during the analysis to define the section behavior,” Section 29.6.5). Because the values given in temperature fields and predefined fields are ignored, the absolute amplitude value will be used to define both the temperature and the gradient and field and gradient, respectively. Input File Usage: \*AMPLITUDE, NAME=name, VALUE=ABSOLUTE Abaqus/CAE Usage: Absolute amplitude magnitudes are not supported in Abaqus/CAE. # Defining the amplitude data The variation of an amplitude with time can be specified in several ways. The variation of an amplitude with frequency can be given only in tabular or equally spaced form. # Defining tabular data Choose the tabular definition method (default) to define the amplitude curve as a table of values at convenient points on the time scale. Abaqus interpolates linearly between these values, as needed. By default in Abaqus/Standard, if the time derivatives of the function must be computed, some smoothing is applied at the time points where the time derivatives are discontinuous. In contrast, in Abaqus/Explicit no default smoothing is applied (other than the inherent smoothing associated with a finite time increment). You can modify the default smoothing values (smoothing is discussed in more detail below, under the heading “Using an amplitude definition with boundary conditions”); alternatively, a smooth step amplitude curve can be defined (see “Defining smooth step data” below). If the amplitude varies rapidly—as with the ground acceleration in an earthquake, for example—you must ensure that the time increment used in the analysis is small enough to pick up the amplitude variation accurately since Abaqus will sample the amplitude definition only at the times corresponding to the increments being used. If the analysis time in a step is less than the earliest time for which data exist in the table, Abaqus applies the earliest value in the table for all step times less than the earliest tabulated time. Similarly, if the analysis continues for step times past the last time for which data are defined in the table, the last value in the table is applied for all subsequent time. Several examples of tabular input are shown in Figure 34.1.2–1. Input File Usage: \*AMPLITUDE, NAME=name, DEFINITION=TABULAR Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: Tabular # Defining equally spaced data Choose the equally spaced definition method to give a list of amplitude values at fixed time intervals beginning at a specified value of time. Abaqus interpolates linearly between each time interval. You must specify the fixed time (or frequency) interval at which the amplitude data will be given, $\Delta t .$ You can also specify the time (or lowest frequency) at which the first amplitude is given, $t _ { 0 } ;$ ; the default is $t _ { 0 } { = } 0 . 0$ . If the analysis time in a step is less than the earliest time for which data exist in the table, Abaqus applies the earliest value in the table for all step times less than the earliest tabulated time. Similarly,

		Amplitude Table:
a. Uniformly increasing load		Time	Relative load
Relative load magnitude	1.0	0.0	0.0
		1.0	1.0
0.0
b. Uniformly decreasing load
Relative load magnitude	1.0	0.0	1.0
		1.0	0.0
0.0
c. Variable load
Relative load magnitude	1.0	0.0	0.0
		0.4	1.2
		0.6	0.5
		0.8	0.5
		1.0	0.0
0.0

Figure 34.1.2–1 Tabular amplitude definition examples. if the analysis continues for step times past the last time for which data are defined in the table, the last value in the table is applied for all subsequent time. Input File Usage: \*AMPLITUDE, NAME=name, DEFINITION=EQUALLY SPACED, FIXED INTERVAL= , BEGIN= Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: Equally spaced: Fixed interval: The time (or lowest frequency) at which the first amplitude is given, $t _ { 0 } ,$ , is indicated in the first table cell. # Defining periodic data Choose the periodic definition method to define the amplitude, a, as a Fourier series: $$ \begin{array}{l} a = A _ {0} + \sum_ {n = 1} ^ {N} \left[ A _ {n} \cos n \omega \left(t - t _ {0}\right) + B _ {n} \sin n \omega \left(t - t _ {0}\right) \right] \quad \text { for } \quad t \geq t _ {0}, \\ a = A _ {0} \quad \text { for } \quad t < t _ {0}, \\ \end{array} $$ where $t _ { 0 } , N , \omega , A _ { 0 } , A _ { n }$ , and $B _ { n } , n = 1 , 2 \ldots N$ , are user-defined constants. An example of this form of input is shown in Figure 34.1.2–2. Input File Usage: \*AMPLITUDE, NAME=name, DEFINITION=PERIODIC Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: Periodic 

line

| Time | a | |-------|-------| | 0.00 | -0.10 | | 0.05 | 0.60 | | 0.10 | -0.40 | | 0.15 | 0.20 | | 0.20 | -0.40 | | 0.25 | 0.60 | | 0.30 | -0.40 | | 0.35 | 0.20 | | 0.40 | -0.40 | | 0.45 | 0.60 | | 0.50 | -0.40 |

$$ p = 0. 2 s $$ $$ a = A _ {0} + \sum_ {n = 1} ^ {N} \left[ A _ {n} \cos n \omega \left(t - t _ {0}\right) + B _ {n} \sin n \omega \left(t - t _ {0}\right) \right] \quad \text { for } t \geq t _ {0} $$ $$ a = A _ {0} \quad \text { for } t < t _ {0} $$ with $$ N = 2, \omega = 3 1. 4 1 6 \mathrm{rad/s}, t _ {0} = - 0. 1 6 1 4 \mathrm{s} $$ $$ \mathrm{A} _ {0} = 0, \mathrm{A} _ {1} = 0. 2 2 7, \mathrm{B} _ {1} = 0. 0, \mathrm{A} _ {2} = 0. 4 1 3, \mathrm{B} _ {2} = 0. 0 $$ Figure 34.1.2–2 Periodic amplitude definition example. # Defining modulated data Choose the modulated definition method to define the amplitude, a, as $$ a = A _ {0} + A \sin \omega_ {1} (t - t _ {0}) \sin \omega_ {2} (t - t _ {0}) \quad \text {for} \quad t > t _ {0}, $$ $$ a = A _ {0} \qquad \text {for} \quad t \leq t _ {0}, $$ where $A _ { 0 } , A , t _ { 0 } , \omega _ { 1 }$ , and $\omega _ { 2 }$ are user-defined constants. An example of this form of input is shown in Figure 34.1.2–3. Input File Usage: \*AMPLITUDE, NAME=name, DEFINITION=MODULATED Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: Modulated 

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| Time (x 10⁻¹) | a | | ------------- | ------- | | 0 | 1.0 | | 2 | 1.0 | | 3 | -1.0 | | 4 | 2.5 | | 5 | -1.0 | | 6 | 2.5 | | 7 | -1.0 | | 8 | 2.5 | | 9 | -1.0 | | 10 | 2.5 |

$$ a = A _ {0} + A \sin \omega_ {1} (t - t _ {0}) \sin \omega_ {2} (t - t _ {0}) \quad \text {for} t > t _ {0} $$ $$ a = A _ {0} \quad \text { for } t \leq t _ {0} $$ with $$ A _ {0} = 1. 0, \quad A = 2. 0, \quad \omega_ {1} = 1 0 \pi , \quad \omega_ {2} = 2 0 \pi , \quad t _ {0} = . 2 $$ Figure 34.1.2–3 Modulated amplitude definition example. # Defining exponential decay Choose the exponential decay definition method to define the amplitude, a, as $$ a = A _ {0} + A \exp (- (t - t _ {0}) / t _ {d}) \quad \text {for} \quad t \geq t _ {0}, $$ $$ a = A _ {0} \quad \text { for } \quad t < t _ {0}, $$ where $A _ { 0 } , A , t _ { 0 }$ , and $t _ { d }$ are user-defined constants. An example of this form of input is shown in Figure 34.1.2–4. Input File Usage: \*AMPLITUDE, NAME=name, DEFINITION=DECAY Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: Decay 

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| Time (x 10⁻¹) | a | | ------------- | ---- | | 0 | 0.0 | | 2 | 5.0 | | 3 | 3.0 | | 4 | 2.0 | | 5 | 1.5 | | 6 | 1.0 | | 7 | 0.7 | | 8 | 0.4 | | 9 | 0.2 | | 10 | 0.1 |

$$ a = A _ {0} + A \exp \left[ - \left(t - t _ {0}\right) / t _ {d} \right] \quad \text { for } t \geq t _ {0} $$ $$ a = A _ {0} \quad \text { for } t < t _ {0} $$ with $$ \mathsf {A} _ {0} = 0. 0, \quad \mathsf {A} = 5. 0, \quad \mathsf {t} _ {0} = 0. 2, \quad \mathsf {t} _ {\mathrm{d}} = 0. 2 $$ Figure 34.1.2–4 Exponential decay amplitude definition example. # Defining smooth step data Abaqus/Standard and Abaqus/Explicit can calculate amplitudes based on smooth step data. Choose the smooth step definition method to define the amplitude, a, between two consecutive data points $( t _ { i } , A _ { i } )$ （204号 and $\left( t _ { i + 1 } , A _ { i + 1 } \right)$ as $$ a = A _ {i} + (A _ {i + 1} - A _ {i}) \xi^ {3} (1 0 - 1 5 \xi + 6 \xi^ {2}) \qquad \mathrm{for} \quad t _ {i} \leq t \leq t _ {i + 1}, $$ where $\xi = ( t - t _ { i } ) / ( t _ { i + 1 } - t _ { i } )$ . The above function is such that $a = A _ { i }$ at $t _ { i } , a = A _ { i + 1 }$ at $t _ { i + 1 }$ , and the first and second derivatives of a are zero at $t _ { i }$ and $t _ { i + 1 }$ . This definition is intended to ramp up or down smoothly from one amplitude value to another. The amplitude, ${ \pmb a } ,$ is defined such that $$ a = A _ {0} \quad \text { for } \quad t \leq t _ {0}, $$ $$ a = A _ {f} \quad \text { for } \quad t \geq t _ {f}, $$ where $( t _ { 0 } , A _ { 0 } )$ and $( t _ { f } , A _ { f } )$ are the first and last data points, respectively. Examples of this form of input are shown in Figure 34.1.2–5 and Figure 34.1.2–6. This definition cannot be used to interpolate smoothly between a set of data points; i.e., this definition cannot be used to do curve fitting. Input File Usage: \*AMPLITUDE, NAME=name, DEFINITION=SMOOTH STEP Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: Smooth step # Defining a solution-dependent amplitude for superplastic forming analysis Abaqus/Standard can calculate amplitude values based on a solution-dependent variable. Choose the solution-dependent definition method to create a solution-dependent amplitude curve. The data consist of an initial value, a minimum value, and a maximum value. The amplitude starts with the initial value and is then modified based on the progress of the solution, subject to the minimum and maximum values. The maximum value is typically the controlling mechanism used to end the analysis. This method is used with creep strain rate control for superplastic forming analysis (see “Rate-dependent plasticity: creep and swelling,” Section 23.2.4). Input File Usage: \*AMPLITUDE, NAME=name, DEFINITION=SOLUTION DEPENDENT Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: Solution dependent # Defining the bubble load amplitude for an underwater explosion Two interfaces are available in Abaqus for applying incident wave loads (see “Incident wave loading due to external sources” in “Acoustic and shock loads,” Section 34.4.6). For either interface bubble dynamics can be described using a model internal to Abaqus. A description of this built-in mechanical model and the parameters that define the bubble behavior are discussed in “Defining bubble loading for spherical incident wave loading” in “Acoustic and shock loads,” Section 34.4.6. The related theoretical details are described in “Loading due to an incident dilatational wave field,” Section 6.3.1 of the Abaqus Theory Guide.